In distributed optimization, a large number of machines alternate between local computations and communication with a coordinating server. Communication, which can be slow and costly, is the main bottleneck in this setting. To reduce this burden and therefore accelerate distributed gradient descent, two strategies are popular: 1) communicate less frequently; that is, perform several iterations of local computations between the communication rounds; and 2) communicate compressed information instead of full-dimensional vectors. We propose CompressedScaffnew, the first algorithm for distributed optimization that jointly harnesses these two strategies and converges linearly to an exact solution in the strongly convex setting, with a doubly accelerated rate: it benefits from the two acceleration mechanisms provided by local training and compression, namely a better dependency on the condition number of the functions and on the dimension of the model, respectively.

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